Question: What is the modulo $5$ remainder of the sum $1+2+3+4+5+ \ldots + 120+121+122+123?$
Answer: Instead of adding up the sum and finding the residue, we can find the residue of each number to make computation easier.

Each group of $5$ numbers would have the sum of residues $1+2+3+4+0=10$. Since $10 \equiv 0 \pmod{5}$, we can ignore every group of $5$.

This leaves the numbers $121,122,$ and $123$. The sum of the residues is $1+2+3 \equiv 6 \equiv \boxed{1} \pmod{5}$.